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 5th Year
Maths
Ordinary Level
David Lewis
Number Systems
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Contents: Number Systems ....................................................................... 3 1. Number types ............................................................................................................................................................................... 3 2. Prime numbers ............................................................................................................................................................................. 7 3. Significant figures ....................................................................................................................................................................... 8 4. Scientific notation ....................................................................................................................................................................... 9 5. Putting the numbers in order ............................................................................................................................................. 11 6. Past and probable exam questions ................................................................................................................................... 12 7. Solutions to Number Systems ............................................................................................................................................. 15 ©The Dublin School of Grinds Page 1 David Lewis Number Systems Number systems is worth 0% to 4% of the Leaving Cert. It appears on Paper 1. 1. Number types After certain questions we often see the following symbols: x ∈ N x ∈ Z x ∈ R x ∈ Q x ∈ R/Q N stands for Natural numbers …or as I say ‘Normal numbers’. Normal numbers are the ones we use in everyday life such as 1, 2, 3, 4, 5, etc. They are positive whole numbers. Note: Despite what some text books say, zero is not considered a natural number by The Irish State Examinations Commission. Example 1 Draw x ≥ 3 on the numberline, x ∈ N Notice that we only fill in the whole numbers and not the bits in between. We also fill in the arrow at the end to show x ≥ 3 goes on forever. Example 2 Draw x > 3 on the numberline, x ∈ N This is the same as eg1. except that we did not fill in the 3 itself, because the question said > 3, not ≥ 3. Comments ©The Dublin School of Grinds Page 3 David Lewis Example 3 Draw x ≤ 4 on the numberline, x ∈ N Z stands for Integers They are positive or negative whole numbers. Examples of integers are: -­‐7, -­‐3. 0, 4, 12 Example 4 Draw x ≤ 3 on the numberline, x ∈ Z Notice that we fill in the arrow at the end to show x ≤ 3 goes on forever. Example 5 Draw x > -­‐4 on the numberline, x ∈ Z Notice that we did not fill in the -­‐4 itself, because the question said > -­‐4, not ≥ -­‐4. R stands for Real numbers …or as I say ‘Random numbers’. Random numbers are any number, (except imaginary numbers which we meet in complex numbers). They can be plus or minus, they can be whole numbers, fraction numbers, decimal numbers….anything!! !
Examples of random numbers are: -­‐9, − , -­‐0.6, 0, 2, 7.3, 9 !
Example 8 Draw x ≥ 2 on the numberline, x ∈ R Notice that we fill in everything from 2 right up to and including the arrow. Comments ©The Dublin School of Grinds Page 4 David Lewis Example 9 Draw x > 2 on the numberline, x ∈ R This is the same as eg6. except that we have not filled in the 2 itself, because the question said > 2, not ≥ 2. We used a hollow circle around the 2 to show we are excluding the 2. Q stands for Rational numbers …or as I say ‘fraction numbers’. These are numbers which can be written as a fraction, for example: ! ! !! !
!
! !
!
, , , , 7 =
!
!
, −12 =
!!"
!
You will not be asked to draw these on the numberline, however you are required to know this at the end of certain questions, as shown in the example below… Example 10 Write !!!" !
!
!
!" !
!
!
= +
!
! !
!
!
= +
as 𝑎 + 𝑏 7 where a, b ∈ Q R/Q stands for Irrational number … or as I say ‘non-­‐fraction numbers’. These are numbers which cannot be written as a fraction, for example: 2, 3, 99, 𝜋 NOTE: We have seen questions where you’re told 𝜋 =
!!
be written as a fraction, i.e. it is irrational. !
, but this is only a rough value of 𝜋 . The actual value of 𝜋 can’t The reason we use R/Q should be understood from our knowledge of the sets chapter. The back slash means “take away the elements in”. So R/Q means: “Real numbers, take away the elements in rational numbers”. If we put this in more simple English it means: “Random numbers but leave out the fraction numbers”. This clearly leaves us with the non-­‐fraction numbers! Comments ©The Dublin School of Grinds Page 5 David Lewis To find if a number is irrational, simply type it into your calculator and try to change it to a fraction by using the relevant button: Casio: button Sharp: button
If it won’t change to a fraction, then it’s irrational! The types of numbers are often demonstrated by the Examiner on the following Venn diagram: As you see, Natural numbers are inside Integers, because Natural numbers are Integers! Then Natural numbers and Integers are inside Rational numbers, because Natural numbers and Integers are Rational numbers! And so on … Comments ©The Dublin School of Grinds Page 6 David Lewis 2. Prime numbers Prime numbers are natural numbers that have two factors only. For example: 5 is a prime number (factors: 1 and 5) 7 is a prime number (factors: 1 and 7) 6 in not a prime number (factors: 1, 2, 3, 6) In your Leaving Cert you may be asked to “express a number as a product of prime numbers”. To do this you keep dividing in the lowest possible prime number until you get to 1. Let’s look at some examples. Example 1 Express 42 as a product of prime numbers. Solution: => 2×3×7 Example 2 Express 60 as a product of prime numbers: Solution: => 2×2×3×5 = 2! ×3×5 Comments ©The Dublin School of Grinds Page 7 David Lewis 3. Significant figures Rule for Significant Figures: Read from the left and start counting significant figures when you encounter the first non-­‐zero digit. 1) All non zero numbers are significant. 613 has three significant figures
123456 has six significant figures 2) Zeros located between non-­‐zero digits are significant. 5004 has four significant figures
602 has three significant figures 6000000000000002 has 16 significant figures! 3) Trailing zeros (those at the end) are significant only if the number contains a decimal point; otherwise they are insignificant. 5.640 has four significant figures
120000.0 has six significant figures 120000 has two significant figures 4) Zeros to left of the first non-­‐zero digit are insignificant. 0.000456 has three significant figures 0.52 s two significant figures 0. 000000000000000000000000000000000052 also has two significant figures! Comments ©The Dublin School of Grinds Page 8 David Lewis 4. Scientific notation Example 1 The diameters of Venus and Saturn are 1.21×104km and 1.21×105km. What is the difference between the diameters of the two planets? Give your answer in the form of 𝑎×10n where 𝑛 ∈ 𝑍 and 1 ≤ 𝑎 < 10. Solutions: When you subtract these numbers on your calculator, and depending on the mode your calculator is in, you will either get 1.089×10! (which is the answer) or 108900. To change 108900 to scientific notation, just count how many places you must move a decimal in from the right hand side so that just one number is remaining to the left hand side. In this case we must move in 5 places from the right hand side. Therefore the answer is 1.08900×10! . You can ignore zeros at the end, so we have 1.089×10! . Example 2 Light travels at a speed of approximately 2.9×105km/sec. How many kilometres will light travel in 8 minutes? Express your answer in the form 𝑎 ×10! , where 𝑛 ∈ 𝑁 and 1 ≤ 𝑎 < 10. Solution: Multiply by 60 to change to minutes, then multiply by 8 to change to 8 minutes: 2.9 × 10! × 60 × 8 = 139200000 = 1.39200000 × 10! = 1.392 × 10! Question 4.1 Given that 𝑥 = 2×10!! and 𝑦 = 7×10!! , evaluate 𝑥 + 8𝑦. Express your answer in the form 𝑎 ×10! , where 𝑛 ∈ 𝑁 and 1 ≤ 𝑎 < 10. Solution: Comments ©The Dublin School of Grinds Page 9 David Lewis Question 4.2 In 1981 the population of Peru was approximately 1.8×107. By 1988 the population had increased by 2.5 million. What would be the approximate population of Peru in 1988? Express your answer in the form 𝑎 ×10! , where 𝑛 ∈ 𝑁 and 1 ≤ 𝑎 < 10. Solution Comments ©The Dublin School of Grinds Page 10 David Lewis 5. Putting the numbers in order A popular Leaving Cert exam question is to ask you to put a list of numbers in order. The numbers will be given in a mixture of decimals, fractions and square roots. Rule: Change all the numbers to decimals using your calculator. Example 1 By putting the smallest number first place the following number in order: Solution Changing to decimals: 10
= 1.428571429 7
2 = 1.414213562 7
= 1.428869017 2 6
(1.19)! = 1.4161 !"
=> The order is 2, (1.19)! , ,
!
!
! !
NOTE: Sometimes the Examiner can ask you to put them in order with the largest first, so watch out for the wording of the question. Question 5.1 By putting the largest number first, place the following numbers in order: NOTE: When you change the numbers to decimals they are usually close to each other (not always though!). If you get numbers that are totally different, this is a hint that you may be wrong, so make sure to check again on your calculator. Comments ©The Dublin School of Grinds Page 11 David Lewis 6. Past and probable exam questions Question 1 ©The Dublin School of Grinds Page 12 David Lewis Question 2 Question 3 ©The Dublin School of Grinds Page 13 David Lewis Question 4 ©The Dublin School of Grinds Page 14 David Lewis 7. Solutions to Number Systems Question 4.1 𝑥 + 8𝑦 = 2 × 10!! + 8(7 ×10!! )
= 7.6 × 10!!
Note: If your calculator says
!"
just press the ‘change’ button to turn it into a decimal.
!"##
Question 4.2 1.8 × 10! + 2,500,000 = 20500000
= 2.050000 ×10!
= 2.05 × 10!
Question 5.1 Changing to decimals: 7
= 1.166666667 6
6
= 1.224744871 2
1.11 ! = 1.2321 1.3456 = 1.16 => The order is 1.11 ! ,
! !
!
, , 1.3456 !
Solutions to past and probable exam questions Question 1 i)
ii)
5 cannot be written as a fraction Question 2 2!" = 16,77,216 = 1.6772116×10! = 1.67×10! (correct to three significant figures) ©The Dublin School of Grinds Page 15 David Lewis Question 3 a)
Note for students: Rational means it can be written as a fraction. Type each part into your calculator and see if it gives it as a fraction. This will tell you if it is rational or not. b) Question 4 a) Prime numbers are natural numbers that have 2 factors only e.g.: 5 or 7 b) => 2652 = 2×2×3×13×17 => 2652 = 2! ×3×13×17 c) 2!" − 1 = 2.305843009×10!" = 2.3×10!" (correct to 2 significant figures) d) There is the 2 in front of the decimal and then the 18 numbers to the right of the decimal. ð There are 19 digits. ©The Dublin School of Grinds Page 16 David Lewis